In a 1998 advertising campaign, Nabisco claimed that every...

In a 1998 advertising campaign, Nabisco claimed that every 18 -ounce bag of Chips Ahoy! cookies contained at least 1000 chocolate chips. Brad Warner and Jim Rutledge (Chance, Vol. 12, No. 1, 1999) tried to verify the claim. The following data represent the number of chips in an 18-ounce bag of Chips Ahoy! based on their study. $$\begin{array}{lllll}1087 & 1098 & 1103 & 1121 & 1132 \\\hline 1185 & 1191 & 1199 & 1200 & 1213 \\\hline 1239 & 1244 & 1247 & 1258 & 1269 \\\hline 1307 & 1325 & 1345 & 1356 & 1363 \\\hline 1135 & 1137 & 1143 & 1154 & 1166 \\\hline 1214 & 1215 & 1219 & 1219 & 1228 \\\hline 1270 & 1279 & 1293 & 1294 & 1295 \\\hline 1377 & 1402 & 1419 & 1440 & 1514\end{array}$$ (a) Use the following normal probability plot to determine if the data could have come from a normal distribution. (FIGURE CAN'T COPY) (b) Determine the mean and standard deviation of the sample data. (c) Using the sample mean and sample standard deviation obtained in part (b) as estimates for the population mean and population standard deviation, respectively, draw a graph of a normal model for the distribution of chips in a bag a Chips Ahoy!. (d) Using the normal model from part (c), find the probability that an 18 -ounce bag of Chips Ahoy! selected at random contains at least 1000 chips. (e) Using the normal model from part (c), determine the proportion of 18 -ounce bags of Chips Ahoy! that contains between 1200 and 1400 chips.

In a 1998 advertising campaign, Nabisco claimed that every 18 -ounce bag of Chips Ahoy! cookies contained at least 1000 chocolate chips. Brad Warner and Jim Rutledge (Chance, Vol. 12, No. 1, 1999) tried to verify the claim. The following data represent the number of chips in an 18-ounce bag of Chips Ahoy! based on their study.
$$\begin{array}{lllll}1087 & 1098 & 1103 & 1121 & 1132 \\\hline 1185 & 1191 & 1199 & 1200 & 1213 \\\hline 1239 & 1244 & 1247 & 1258 & 1269 \\\hline 1307 & 1325 & 1345 & 1356 & 1363 \\\hline 1135 & 1137 & 1143 & 1154 & 1166 \\\hline 1214 & 1215 & 1219 & 1219 & 1228 \\\hline 1270 & 1279 & 1293 & 1294 & 1295 \\\hline 1377 & 1402 & 1419 & 1440 & 1514\end{array}$$
(a) Use the following normal probability plot to determine if the data could have come from a normal distribution. (FIGURE CAN'T COPY)
(b) Determine the mean and standard deviation of the sample data.
(c) Using the sample mean and sample standard deviation obtained in part (b) as estimates for the population mean and population standard deviation, respectively, draw a graph of a normal model for the distribution of chips in a bag a Chips Ahoy!.
(d) Using the normal model from part (c), find the probability that an 18 -ounce bag of Chips Ahoy! selected at random contains at least 1000 chips.
(e) Using the normal model from part (c), determine the proportion of 18 -ounce bags of Chips Ahoy! that contains between 1200 and 1400 chips.

Key Concepts

Standardization (Z-scores)

Standardization is the process of converting a normal random variable to a standard normal variable using the z-score. This involves subtracting the mean from a data value and dividing by the standard deviation. Z-scores allow for the comparison of data points from different normal distributions by converting them into a common scale, facilitating the computation of probabilities using standard normal tables.

Probability Calculations

Probability calculations using the normal model involve determining the area under the normal curve corresponding to specific intervals or tail regions. These calculations often use z-scores and standard normal distribution tables (or software) to estimate the likelihood of observing a value or range of values within a normally distributed dataset. This method is essential for tasks such as hypothesis testing and confidence interval estimation.

Sample Statistics

Sample statistics, such as the sample mean and sample standard deviation, are estimates of the population parameters based on observed data. The sample mean provides a measure of central tendency, while the sample standard deviation quantifies the spread or variability of the data. These statistics serve as the building blocks for inferential procedures, including the construction of a normal model.

Normal Probability Plot

A normal probability plot, also known as a QQ plot, is a graphical tool used to assess whether a dataset approximates a normal distribution. By plotting the quantiles of the sample data against the theoretical quantiles of a normal distribution, one can visually determine deviations from normality; if the plot shows points falling approximately along a straight line, the normality assumption is reasonable.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell shape, and its behavior is fully determined by the mean and standard deviation. This distribution is widely used in statistics for modeling naturally occurring phenomena.

Transcript

00:01 Alright, so we're looking at the number of chips and a chocolate chips in a bag of cookies.

00:07 So that's what we're looking at.

00:09 That's the data here.

00:14 And i sorted it.

00:16 I put it in a big column and sorted it to help me just see it better.

00:21 And there's a normal probability plot that was created for us, and we're asking if this shows that it's normally distributed.

00:29 And yes, this does.

00:30 I recreated it here as a normal probability.

00:32 Like sketch and then the way a lot of these some of these programs use or create probability plots as they make this this little parameter here this range on the top and the range on the bottom here of this line and if your data points fall within that range then it's assumed that it's a normal or it's a way you can justify it being normally distributed so the answer is yes we can use the normal this probability probability plot does show the data could have come from a normal distribution.

01:05 Then we're told to compute the sample mean and sample standard deviation, and now i've used a spreadsheet function for that.

01:10 Go spreadsheets.

01:12 If you haven't had a chance to play spreadsheets yet, my goodness, they're just awesome.

01:16 So those i put over, oh, here they are.

01:22 The mean of the dataset is this many, 1 ,247 .375 chips.

01:27 The standard deviation is about 100 and then if we assume those to be the population parameters we want to draw a graph of a normal model for the distribution and so for that we need z scores actually no actually we could do this without it if we just do a regular normal non standardized distribution normal distribution with the population mean and standard deviation.

02:03 Then we can do this and i'll draw the picture over here.

02:06 These pictures just help so much.

02:09 Pictures are great.

02:12 So here's our normal distribution.

02:15 Shouldn't have that flare.

02:16 Hang on.

02:17 I'm assuming it would be like this.

02:18 Something like that.

02:20 And the mean is right here.

02:21 And the mean is 1247.

02:27 That's 2 .4.

02:28 And then the sample standard deviation is plus 100.

02:35 So that's what i kind of recreated here in tabular in a spreadsheet form.

02:40 So there's the mean.

02:41 And then one standard deviation is roughly, i'm just going to say plus 101.

02:52 I'm going to round that to be 101.

02:55 So then would be, what would that be 1347, roughly 1348 here.

03:02 So there's one standard deviation, 48 .4.

03:10 I'm approximating here.

03:14 The staring deviation is to quite a few levels of precision, so i'm just going to ignore that.

03:20 And then we add another 110, or 101, excuse me, to get the second stern deviation, which is this number, 14 .49 .3.

03:35 Well, if we're using the 101, we should probably just be consistent here.

03:40 So that's going to be, what, 1449 .4.

03:44 Like that and then the third standard deviation which i'll do here is going to be what they say 1550 point well we're going to do we're adding we're assuming it to be 101 just for simplicity's sake because that about you i count chocolate chips and hold chips maybe you could say half of a chip but 0 .96 of a chip i'm not gonna i'm going to assume that to be a whole chip so that's why i'm going to go with that one.

04:23 And then you also have the negative values like this, you know, my or subtracting the standard deviation and then we get 1146, assuming that to be the mean.

04:48 Let's see, then what else do we get? we get, i did blue, so we're going to subtract another 101.

04:57 So we have these nice little like little bounds here for our standard deviation.

05:04 So one standard deviation away, two standard deviations away.

05:08 We get 1045 .4.

05:13 And then last one, shoot.

05:16 We get subtracted another one.

05:18 And there's three standard deviations away, which is 944 .4 .4.

05:24 So there you go.

05:25 And that is how that works.

05:28 There's our picture of the normal distribution, non -standardized.

05:39 Using, let's see, we did that for a part b, there we go, there's that, oh again, for part, this is part c that we just did.

05:44 Part d, use the normal model to find the probability that an 18 -ounce bag of the cookies has at least 1 ,000 chips.

05:55 So this is why this picture is kind of nice, because we can kind of make sense of this a little bit more with this value.

06:01 I'll do this in green because that's the one we go.

06:02 I want to go ahead and figure out.

06:07 And we are going to find a thousand.

06:11 So a thousand is, here's 1 ,000.

06:14 So here's, i'm going to just sketch in here.

06:17 Here's 1 ,000.

06:21 And we want to find at least that.

06:25 So we want to have this.

06:29 Let me make sure i'm reading that right.

06:31 It says at least 1 ,000 ships.

06:34 So 1 ,000 or more.

06:35 So basically we want all of this stuff.

06:38 We want all of this.

06:39 My drawing is getting all covered.

06:43 We want all of this.

06:47 That's what we want.

06:52 Well, how do we get that? well, we can use the z score for, if we standardize it, and then here's the mu, which would be zero.

07:04 Let's put zero here.

07:07 Zero.

07:08 And then if we subtract standard deviations, we can see here's one, two, we're going to be a little more than two standard deviations away...

Please give Ace some feedback